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The boundary-Wecken classification of surfaces

Robert F Brown and Michael R Kelly

Algebraic & Geometric Topology 4 (2004) 49–71

arXiv: math.AT/0402334


Let X be a compact 2-manifold with nonempty boundary X and let f : (X,X) (X,X) be a boundary-preserving map. Denote by MF[f] the minimum number of fixed point among all boundary-preserving maps that are homotopic through boundary-preserving maps to f. The relative Nielsen number N(f) is the sum of the number of essential fixed point classes of the restriction f̄: X X and the number of essential fixed point classes of f that do not contain essential fixed point classes of f̄. We prove that if X is the Möbius band with one (open) disc removed, then MF[f] N(f) 1 for all maps f : (X,X) (X,X). This result is the final step in the boundary-Wecken classification of surfaces, which is as follows. If X is the disc, annulus or Möbius band, then X is boundary-Wecken, that is, MF[f] = N(f) for all boundary-preserving maps. If X is the disc with two discs removed or the Möbius band with one disc removed, then X is not boundary-Wecken, but MF[f] N(f) 1. All other surfaces are totally non-boundary-Wecken, that is, given an integer k 1, there is a map fk: (X,X) (X,X) such that MF[fk] N(fk) k.

boundary-Wecken, relative Nielsen number, punctured Möbius band, boundary-preserving map
Mathematical Subject Classification 2000
Primary: 55M20
Secondary: 54H25, 57N05
Forward citations
Received: 21 November 2002
Revised: 15 October 2003
Accepted: 26 November 2003
Published: 7 February 2004
Robert F Brown
Department of Mathematics
University of California
Los Angeles CA 90095-1555
Michael R Kelly
Department of Mathematics and Computer Science
Loyola University
6363 St Charles Avenue
New Orleans LA 70118